Vol. 11, No. 1, 2018

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Continuous dependence and differentiating solutions of a second order boundary value problem with average value condition

Jeffrey W. Lyons, Samantha A. Major and Kaitlyn B. Seabrook

Vol. 11 (2018), No. 1, 95–102
DOI: 10.2140/involve.2018.11.95
Abstract

Using a few conditions, continuous dependence, and a result regarding smoothness of initial conditions, we show that derivatives of solutions to the second order boundary value problem y = f(x,y,y), a < x < b, satisfying y(x1) = y1, 1(d c)cdy(x)x̣ = y2, where a < x1 < c < d < b and y1,y2 with respect to each of the boundary data x1, y1, y2, c, d solve the associated variational equation with interesting boundary conditions. Of note is the second boundary condition, which is an average value condition.

Keywords
continuous dependence, boundary data smoothness, average value condition, Peano's theorem
Mathematical Subject Classification 2010
Primary: 34B10
Milestones
Received: 3 August 2016
Revised: 13 August 2016
Accepted: 28 August 2016
Published: 17 July 2017

Communicated by Martin J. Bohner
Authors
Jeffrey W. Lyons
Department of Mathematics
Nova Southeastern University
3301 College Ave
Fort Lauderdale, FL 33314
United States
Samantha A. Major
Department of Mathematics
Nova Southeastern University
3301 College Ave
Fort Lauderdale, FL 33314
United States
Kaitlyn B. Seabrook
Department of Mathematics
Nova Southeastern University
3301 College Ave
Fort Lauderdale, FL 33314
United States